# Properties

 Label 2304.2.d.e.1153.1 Level $2304$ Weight $2$ Character 2304.1153 Analytic conductor $18.398$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.3975326257$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1152) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1153.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2304.1153 Dual form 2304.2.d.e.1153.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000i q^{5} -2.00000 q^{7} +O(q^{10})$$ $$q-2.00000i q^{5} -2.00000 q^{7} -4.00000i q^{11} -2.00000i q^{13} -4.00000 q^{17} +4.00000i q^{19} -8.00000 q^{23} +1.00000 q^{25} +6.00000i q^{29} -6.00000 q^{31} +4.00000i q^{35} -2.00000i q^{37} +12.0000 q^{41} +12.0000i q^{43} +8.00000 q^{47} -3.00000 q^{49} -6.00000i q^{53} -8.00000 q^{55} -8.00000i q^{59} +10.0000i q^{61} -4.00000 q^{65} +8.00000i q^{67} -2.00000 q^{73} +8.00000i q^{77} -14.0000 q^{79} +12.0000i q^{83} +8.00000i q^{85} -8.00000 q^{89} +4.00000i q^{91} +8.00000 q^{95} -2.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{7} + O(q^{10})$$ $$2q - 4q^{7} - 8q^{17} - 16q^{23} + 2q^{25} - 12q^{31} + 24q^{41} + 16q^{47} - 6q^{49} - 16q^{55} - 8q^{65} - 4q^{73} - 28q^{79} - 16q^{89} + 16q^{95} - 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ − 2.00000i − 0.894427i −0.894427 0.447214i $$-0.852416\pi$$
0.894427 0.447214i $$-0.147584\pi$$
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 4.00000i − 1.20605i −0.797724 0.603023i $$-0.793963\pi$$
0.797724 0.603023i $$-0.206037\pi$$
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ 4.00000i 0.917663i 0.888523 + 0.458831i $$0.151732\pi$$
−0.888523 + 0.458831i $$0.848268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.00000i 1.11417i 0.830455 + 0.557086i $$0.188081\pi$$
−0.830455 + 0.557086i $$0.811919\pi$$
$$30$$ 0 0
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 4.00000i 0.676123i
$$36$$ 0 0
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 12.0000 1.87409 0.937043 0.349215i $$-0.113552\pi$$
0.937043 + 0.349215i $$0.113552\pi$$
$$42$$ 0 0
$$43$$ 12.0000i 1.82998i 0.403473 + 0.914991i $$0.367803\pi$$
−0.403473 + 0.914991i $$0.632197\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 0 0
$$55$$ −8.00000 −1.07872
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 8.00000i − 1.04151i −0.853706 0.520756i $$-0.825650\pi$$
0.853706 0.520756i $$-0.174350\pi$$
$$60$$ 0 0
$$61$$ 10.0000i 1.28037i 0.768221 + 0.640184i $$0.221142\pi$$
−0.768221 + 0.640184i $$0.778858\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −4.00000 −0.496139
$$66$$ 0 0
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 8.00000i 0.911685i
$$78$$ 0 0
$$79$$ −14.0000 −1.57512 −0.787562 0.616236i $$-0.788657\pi$$
−0.787562 + 0.616236i $$0.788657\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 0 0
$$85$$ 8.00000i 0.867722i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −8.00000 −0.847998 −0.423999 0.905663i $$-0.639374\pi$$
−0.423999 + 0.905663i $$0.639374\pi$$
$$90$$ 0 0
$$91$$ 4.00000i 0.419314i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 8.00000 0.820783
$$96$$ 0 0
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 6.00000i − 0.597022i −0.954406 0.298511i $$-0.903510\pi$$
0.954406 0.298511i $$-0.0964900\pi$$
$$102$$ 0 0
$$103$$ −14.0000 −1.37946 −0.689730 0.724066i $$-0.742271\pi$$
−0.689730 + 0.724066i $$0.742271\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 8.00000i 0.773389i 0.922208 + 0.386695i $$0.126383\pi$$
−0.922208 + 0.386695i $$0.873617\pi$$
$$108$$ 0 0
$$109$$ − 18.0000i − 1.72409i −0.506834 0.862044i $$-0.669184\pi$$
0.506834 0.862044i $$-0.330816\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −8.00000 −0.752577 −0.376288 0.926503i $$-0.622800\pi$$
−0.376288 + 0.926503i $$0.622800\pi$$
$$114$$ 0 0
$$115$$ 16.0000i 1.49201i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 12.0000i − 1.07331i
$$126$$ 0 0
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ − 8.00000i − 0.693688i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 4.00000 0.341743 0.170872 0.985293i $$-0.445342\pi$$
0.170872 + 0.985293i $$0.445342\pi$$
$$138$$ 0 0
$$139$$ 8.00000i 0.678551i 0.940687 + 0.339276i $$0.110182\pi$$
−0.940687 + 0.339276i $$0.889818\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ 12.0000 0.996546
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 6.00000i 0.491539i 0.969328 + 0.245770i $$0.0790407\pi$$
−0.969328 + 0.245770i $$0.920959\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 12.0000i 0.963863i
$$156$$ 0 0
$$157$$ 18.0000i 1.43656i 0.695756 + 0.718278i $$0.255069\pi$$
−0.695756 + 0.718278i $$0.744931\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 0 0
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 0 0
$$175$$ −2.00000 −0.151186
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 24.0000i − 1.79384i −0.442189 0.896922i $$-0.645798\pi$$
0.442189 0.896922i $$-0.354202\pi$$
$$180$$ 0 0
$$181$$ 2.00000i 0.148659i 0.997234 + 0.0743294i $$0.0236816\pi$$
−0.997234 + 0.0743294i $$0.976318\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −4.00000 −0.294086
$$186$$ 0 0
$$187$$ 16.0000i 1.17004i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 0 0
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 26.0000i 1.85242i 0.377004 + 0.926212i $$0.376954\pi$$
−0.377004 + 0.926212i $$0.623046\pi$$
$$198$$ 0 0
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 12.0000i − 0.842235i
$$204$$ 0 0
$$205$$ − 24.0000i − 1.67623i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ − 8.00000i − 0.550743i −0.961338 0.275371i $$-0.911199\pi$$
0.961338 0.275371i $$-0.0888008\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 24.0000 1.63679
$$216$$ 0 0
$$217$$ 12.0000 0.814613
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 8.00000i 0.538138i
$$222$$ 0 0
$$223$$ −18.0000 −1.20537 −0.602685 0.797980i $$-0.705902\pi$$
−0.602685 + 0.797980i $$0.705902\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 0 0
$$229$$ − 14.0000i − 0.925146i −0.886581 0.462573i $$-0.846926\pi$$
0.886581 0.462573i $$-0.153074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −24.0000 −1.57229 −0.786146 0.618041i $$-0.787927\pi$$
−0.786146 + 0.618041i $$0.787927\pi$$
$$234$$ 0 0
$$235$$ − 16.0000i − 1.04372i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 6.00000i 0.383326i
$$246$$ 0 0
$$247$$ 8.00000 0.509028
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 12.0000i 0.757433i 0.925513 + 0.378717i $$0.123635\pi$$
−0.925513 + 0.378717i $$0.876365\pi$$
$$252$$ 0 0
$$253$$ 32.0000i 2.01182i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 4.00000i 0.248548i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −12.0000 −0.737154
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 26.0000i − 1.58525i −0.609711 0.792624i $$-0.708714\pi$$
0.609711 0.792624i $$-0.291286\pi$$
$$270$$ 0 0
$$271$$ 14.0000 0.850439 0.425220 0.905090i $$-0.360197\pi$$
0.425220 + 0.905090i $$0.360197\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 4.00000i − 0.241209i
$$276$$ 0 0
$$277$$ 18.0000i 1.08152i 0.841178 + 0.540758i $$0.181862\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −24.0000 −1.41668
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 22.0000i − 1.28525i −0.766179 0.642627i $$-0.777845\pi$$
0.766179 0.642627i $$-0.222155\pi$$
$$294$$ 0 0
$$295$$ −16.0000 −0.931556
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 16.0000i 0.925304i
$$300$$ 0 0
$$301$$ − 24.0000i − 1.38334i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 20.0000 1.14520
$$306$$ 0 0
$$307$$ − 24.0000i − 1.36975i −0.728659 0.684876i $$-0.759856\pi$$
0.728659 0.684876i $$-0.240144\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −16.0000 −0.907277 −0.453638 0.891186i $$-0.649874\pi$$
−0.453638 + 0.891186i $$0.649874\pi$$
$$312$$ 0 0
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 0 0
$$319$$ 24.0000 1.34374
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 16.0000i − 0.890264i
$$324$$ 0 0
$$325$$ − 2.00000i − 0.110940i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ − 16.0000i − 0.879440i −0.898135 0.439720i $$-0.855078\pi$$
0.898135 0.439720i $$-0.144922\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 16.0000 0.874173
$$336$$ 0 0
$$337$$ 10.0000 0.544735 0.272367 0.962193i $$-0.412193\pi$$
0.272367 + 0.962193i $$0.412193\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 24.0000i 1.29967i
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ 26.0000i 1.39175i 0.718164 + 0.695874i $$0.244983\pi$$
−0.718164 + 0.695874i $$0.755017\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 4.00000i 0.209370i
$$366$$ 0 0
$$367$$ 10.0000 0.521996 0.260998 0.965339i $$-0.415948\pi$$
0.260998 + 0.965339i $$0.415948\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 12.0000i 0.623009i
$$372$$ 0 0
$$373$$ 22.0000i 1.13912i 0.821951 + 0.569558i $$0.192886\pi$$
−0.821951 + 0.569558i $$0.807114\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ − 20.0000i − 1.02733i −0.857991 0.513665i $$-0.828287\pi$$
0.857991 0.513665i $$-0.171713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ 16.0000 0.815436
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 14.0000i 0.709828i 0.934899 + 0.354914i $$0.115490\pi$$
−0.934899 + 0.354914i $$0.884510\pi$$
$$390$$ 0 0
$$391$$ 32.0000 1.61831
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 28.0000i 1.40883i
$$396$$ 0 0
$$397$$ − 30.0000i − 1.50566i −0.658217 0.752828i $$-0.728689\pi$$
0.658217 0.752828i $$-0.271311\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ 12.0000i 0.597763i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −8.00000 −0.396545
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 16.0000i 0.787309i
$$414$$ 0 0
$$415$$ 24.0000 1.17811
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 20.0000i 0.977064i 0.872546 + 0.488532i $$0.162467\pi$$
−0.872546 + 0.488532i $$0.837533\pi$$
$$420$$ 0 0
$$421$$ − 14.0000i − 0.682318i −0.940006 0.341159i $$-0.889181\pi$$
0.940006 0.341159i $$-0.110819\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −4.00000 −0.194029
$$426$$ 0 0
$$427$$ − 20.0000i − 0.967868i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ −30.0000 −1.44171 −0.720854 0.693087i $$-0.756250\pi$$
−0.720854 + 0.693087i $$0.756250\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 32.0000i − 1.53077i
$$438$$ 0 0
$$439$$ −6.00000 −0.286364 −0.143182 0.989696i $$-0.545733\pi$$
−0.143182 + 0.989696i $$0.545733\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 12.0000i − 0.570137i −0.958507 0.285069i $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ 0 0
$$445$$ 16.0000i 0.758473i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 28.0000 1.32140 0.660701 0.750649i $$-0.270259\pi$$
0.660701 + 0.750649i $$0.270259\pi$$
$$450$$ 0 0
$$451$$ − 48.0000i − 2.26023i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 8.00000 0.375046
$$456$$ 0 0
$$457$$ −6.00000 −0.280668 −0.140334 0.990104i $$-0.544818\pi$$
−0.140334 + 0.990104i $$0.544818\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 34.0000i 1.58354i 0.610821 + 0.791769i $$0.290840\pi$$
−0.610821 + 0.791769i $$0.709160\pi$$
$$462$$ 0 0
$$463$$ −2.00000 −0.0929479 −0.0464739 0.998920i $$-0.514798\pi$$
−0.0464739 + 0.998920i $$0.514798\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 4.00000i 0.185098i 0.995708 + 0.0925490i $$0.0295015\pi$$
−0.995708 + 0.0925490i $$0.970499\pi$$
$$468$$ 0 0
$$469$$ − 16.0000i − 0.738811i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 48.0000 2.20704
$$474$$ 0 0
$$475$$ 4.00000i 0.183533i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4.00000i 0.181631i
$$486$$ 0 0
$$487$$ 38.0000 1.72194 0.860972 0.508652i $$-0.169856\pi$$
0.860972 + 0.508652i $$0.169856\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ − 32.0000i − 1.44414i −0.691820 0.722070i $$-0.743191\pi$$
0.691820 0.722070i $$-0.256809\pi$$
$$492$$ 0 0
$$493$$ − 24.0000i − 1.08091i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ − 32.0000i − 1.43252i −0.697835 0.716258i $$-0.745853\pi$$
0.697835 0.716258i $$-0.254147\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ −12.0000 −0.533993
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 2.00000i − 0.0886484i −0.999017 0.0443242i $$-0.985887\pi$$
0.999017 0.0443242i $$-0.0141135\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 28.0000i 1.23383i
$$516$$ 0 0
$$517$$ − 32.0000i − 1.40736i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −4.00000 −0.175243 −0.0876216 0.996154i $$-0.527927\pi$$
−0.0876216 + 0.996154i $$0.527927\pi$$
$$522$$ 0 0
$$523$$ − 4.00000i − 0.174908i −0.996169 0.0874539i $$-0.972127\pi$$
0.996169 0.0874539i $$-0.0278730\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 24.0000 1.04546
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 24.0000i − 1.03956i
$$534$$ 0 0
$$535$$ 16.0000 0.691740
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 12.0000i 0.516877i
$$540$$ 0 0
$$541$$ 38.0000i 1.63375i 0.576816 + 0.816874i $$0.304295\pi$$
−0.576816 + 0.816874i $$0.695705\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −36.0000 −1.54207
$$546$$ 0 0
$$547$$ − 4.00000i − 0.171028i −0.996337 0.0855138i $$-0.972747\pi$$
0.996337 0.0855138i $$-0.0272532\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ 28.0000 1.19068
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 2.00000i 0.0847427i 0.999102 + 0.0423714i $$0.0134913\pi$$
−0.999102 + 0.0423714i $$0.986509\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 36.0000i 1.51722i 0.651546 + 0.758610i $$0.274121\pi$$
−0.651546 + 0.758610i $$0.725879\pi$$
$$564$$ 0 0
$$565$$ 16.0000i 0.673125i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −4.00000 −0.167689 −0.0838444 0.996479i $$-0.526720\pi$$
−0.0838444 + 0.996479i $$0.526720\pi$$
$$570$$ 0 0
$$571$$ − 40.0000i − 1.67395i −0.547243 0.836974i $$-0.684323\pi$$
0.547243 0.836974i $$-0.315677\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −8.00000 −0.333623
$$576$$ 0 0
$$577$$ −10.0000 −0.416305 −0.208153 0.978096i $$-0.566745\pi$$
−0.208153 + 0.978096i $$0.566745\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 24.0000i − 0.995688i
$$582$$ 0 0
$$583$$ −24.0000 −0.993978
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 32.0000i − 1.32078i −0.750922 0.660391i $$-0.770391\pi$$
0.750922 0.660391i $$-0.229609\pi$$
$$588$$ 0 0
$$589$$ − 24.0000i − 0.988903i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −24.0000 −0.985562 −0.492781 0.870153i $$-0.664020\pi$$
−0.492781 + 0.870153i $$0.664020\pi$$
$$594$$ 0 0
$$595$$ − 16.0000i − 0.655936i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 18.0000 0.734235 0.367118 0.930175i $$-0.380345\pi$$
0.367118 + 0.930175i $$0.380345\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 10.0000i 0.406558i
$$606$$ 0 0
$$607$$ 14.0000 0.568242 0.284121 0.958788i $$-0.408298\pi$$
0.284121 + 0.958788i $$0.408298\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 16.0000i − 0.647291i
$$612$$ 0 0
$$613$$ 22.0000i 0.888572i 0.895885 + 0.444286i $$0.146543\pi$$
−0.895885 + 0.444286i $$0.853457\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 16.0000 0.644136 0.322068 0.946717i $$-0.395622\pi$$
0.322068 + 0.946717i $$0.395622\pi$$
$$618$$ 0 0
$$619$$ 16.0000i 0.643094i 0.946894 + 0.321547i $$0.104203\pi$$
−0.946894 + 0.321547i $$0.895797\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 16.0000 0.641026
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 8.00000i 0.318981i
$$630$$ 0 0
$$631$$ 38.0000 1.51276 0.756378 0.654135i $$-0.226967\pi$$
0.756378 + 0.654135i $$0.226967\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 4.00000i 0.158735i
$$636$$ 0 0
$$637$$ 6.00000i 0.237729i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −36.0000 −1.42191 −0.710957 0.703235i $$-0.751738\pi$$
−0.710957 + 0.703235i $$0.751738\pi$$
$$642$$ 0 0
$$643$$ 12.0000i 0.473234i 0.971603 + 0.236617i $$0.0760386\pi$$
−0.971603 + 0.236617i $$0.923961\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 0 0
$$649$$ −32.0000 −1.25611
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 48.0000i − 1.86981i −0.354892 0.934907i $$-0.615482\pi$$
0.354892 0.934907i $$-0.384518\pi$$
$$660$$ 0 0
$$661$$ 22.0000i 0.855701i 0.903850 + 0.427850i $$0.140729\pi$$
−0.903850 + 0.427850i $$0.859271\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −16.0000 −0.620453
$$666$$ 0 0
$$667$$ − 48.0000i − 1.85857i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 40.0000 1.54418
$$672$$ 0 0
$$673$$ −14.0000 −0.539660 −0.269830 0.962908i $$-0.586968\pi$$
−0.269830 + 0.962908i $$0.586968\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 26.0000i − 0.999261i −0.866239 0.499631i $$-0.833469\pi$$
0.866239 0.499631i $$-0.166531\pi$$
$$678$$ 0 0
$$679$$ 4.00000 0.153506
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 36.0000i − 1.37750i −0.724998 0.688751i $$-0.758159\pi$$
0.724998 0.688751i $$-0.241841\pi$$
$$684$$ 0 0
$$685$$ − 8.00000i − 0.305664i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 36.0000i 1.36950i 0.728776 + 0.684752i $$0.240090\pi$$
−0.728776 + 0.684752i $$0.759910\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 16.0000 0.606915
$$696$$ 0 0
$$697$$ −48.0000 −1.81813
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 10.0000i − 0.377695i −0.982006 0.188847i $$-0.939525\pi$$
0.982006 0.188847i $$-0.0604752\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 12.0000i 0.451306i
$$708$$ 0 0
$$709$$ 2.00000i 0.0751116i 0.999295 + 0.0375558i $$0.0119572\pi$$
−0.999295 + 0.0375558i $$0.988043\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 48.0000 1.79761
$$714$$ 0 0
$$715$$ 16.0000i 0.598366i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −48.0000 −1.79010 −0.895049 0.445968i $$-0.852860\pi$$
−0.895049 + 0.445968i $$0.852860\pi$$
$$720$$ 0 0
$$721$$ 28.0000 1.04277
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 6.00000i 0.222834i
$$726$$ 0 0
$$727$$ −18.0000 −0.667583 −0.333792 0.942647i $$-0.608328\pi$$
−0.333792 + 0.942647i $$0.608328\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 48.0000i − 1.77534i
$$732$$ 0 0
$$733$$ 6.00000i 0.221615i 0.993842 + 0.110808i $$0.0353437\pi$$
−0.993842 + 0.110808i $$0.964656\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 32.0000 1.17874
$$738$$ 0 0
$$739$$ − 24.0000i − 0.882854i −0.897297 0.441427i $$-0.854472\pi$$
0.897297 0.441427i $$-0.145528\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −40.0000 −1.46746 −0.733729 0.679442i $$-0.762222\pi$$
−0.733729 + 0.679442i $$0.762222\pi$$
$$744$$ 0 0
$$745$$ 12.0000 0.439646
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 16.0000i − 0.584627i
$$750$$ 0 0
$$751$$ 26.0000 0.948753 0.474377 0.880322i $$-0.342673\pi$$
0.474377 + 0.880322i $$0.342673\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 20.0000i 0.727875i
$$756$$ 0 0
$$757$$ 2.00000i 0.0726912i 0.999339 + 0.0363456i $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 4.00000 0.145000 0.0724999 0.997368i $$-0.476902\pi$$
0.0724999 + 0.997368i $$0.476902\pi$$
$$762$$ 0 0
$$763$$ 36.0000i 1.30329i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −16.0000 −0.577727
$$768$$ 0 0
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 6.00000i − 0.215805i −0.994161 0.107903i $$-0.965587\pi$$
0.994161 0.107903i $$-0.0344134\pi$$
$$774$$ 0 0
$$775$$ −6.00000 −0.215526
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 48.0000i 1.71978i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 36.0000 1.28490
$$786$$ 0 0
$$787$$ − 20.0000i − 0.712923i −0.934310 0.356462i $$-0.883983\pi$$
0.934310 0.356462i $$-0.116017\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 16.0000 0.568895
$$792$$ 0 0
$$793$$ 20.0000 0.710221
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 42.0000i 1.48772i 0.668338 + 0.743858i $$0.267006\pi$$
−0.668338 + 0.743858i $$0.732994\pi$$
$$798$$ 0 0
$$799$$ −32.0000 −1.13208
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 8.00000i 0.282314i
$$804$$ 0 0
$$805$$ − 32.0000i − 1.12785i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −12.0000 −0.421898 −0.210949 0.977497i $$-0.567655\pi$$
−0.210949 + 0.977497i $$0.567655\pi$$
$$810$$ 0 0
$$811$$ − 12.0000i − 0.421377i −0.977553 0.210688i $$-0.932429\pi$$
0.977553 0.210688i $$-0.0675706\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 8.00000 0.280228
$$816$$ 0 0
$$817$$ −48.0000 −1.67931
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 2.00000i 0.0698005i 0.999391 + 0.0349002i $$0.0111113\pi$$
−0.999391 + 0.0349002i $$0.988889\pi$$
$$822$$ 0 0
$$823$$ −26.0000 −0.906303 −0.453152 0.891434i $$-0.649700\pi$$
−0.453152 + 0.891434i $$0.649700\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 8.00000i 0.278187i 0.990279 + 0.139094i $$0.0444189\pi$$
−0.990279 + 0.139094i $$0.955581\pi$$
$$828$$ 0 0
$$829$$ 14.0000i 0.486240i 0.969996 + 0.243120i $$0.0781709\pi$$
−0.969996 + 0.243120i $$0.921829\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 12.0000 0.415775
$$834$$ 0 0
$$835$$ 16.0000i 0.553703i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −7.00000 −0.241379
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 18.0000i − 0.619219i
$$846$$ 0 0
$$847$$ 10.0000 0.343604
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 16.0000i 0.548473i
$$852$$ 0 0
$$853$$ − 10.0000i − 0.342393i −0.985237 0.171197i $$-0.945237\pi$$
0.985237 0.171197i $$-0.0547634\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −36.0000 −1.22974 −0.614868 0.788630i $$-0.710791\pi$$
−0.614868 + 0.788630i $$0.710791\pi$$
$$858$$ 0 0
$$859$$ 4.00000i 0.136478i 0.997669 + 0.0682391i $$0.0217381\pi$$
−0.997669 + 0.0682391i $$0.978262\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 8.00000 0.272323 0.136162 0.990687i $$-0.456523\pi$$
0.136162 + 0.990687i $$0.456523\pi$$
$$864$$ 0 0
$$865$$ −12.0000 −0.408012
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 56.0000i 1.89967i
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 24.0000i 0.811348i
$$876$$ 0 0
$$877$$ 10.0000i 0.337676i 0.985644 + 0.168838i $$0.0540015\pi$$
−0.985644 + 0.168838i $$0.945999\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −32.0000 −1.07811 −0.539054 0.842271i $$-0.681218\pi$$
−0.539054 + 0.842271i $$0.681218\pi$$
$$882$$ 0 0
$$883$$ 20.0000i 0.673054i 0.941674 + 0.336527i $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 48.0000 1.61168 0.805841 0.592132i $$-0.201714\pi$$
0.805841 + 0.592132i $$0.201714\pi$$
$$888$$ 0 0
$$889$$ 4.00000 0.134156
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 32.0000i 1.07084i
$$894$$ 0 0
$$895$$ −48.0000 −1.60446
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 36.0000i − 1.20067i
$$900$$ 0 0
$$901$$ 24.0000i 0.799556i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 4.00000 0.132964
$$906$$ 0 0
$$907$$ 20.0000i 0.664089i 0.943264 + 0.332045i $$0.107738\pi$$
−0.943264 + 0.332045i $$0.892262\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 0 0
$$913$$ 48.0000 1.58857
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 6.00000 0.197922 0.0989609 0.995091i $$-0.468448\pi$$
0.0989609 + 0.995091i $$0.468448\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 2.00000i − 0.0657596i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 36.0000 1.18112 0.590561 0.806993i $$-0.298907\pi$$
0.590561 + 0.806993i $$0.298907\pi$$
$$930$$ 0 0
$$931$$ − 12.0000i − 0.393284i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 32.0000 1.04651
$$936$$ 0 0
$$937$$ 38.0000 1.24141 0.620703 0.784046i $$-0.286847\pi$$
0.620703 + 0.784046i $$0.286847\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 14.0000i − 0.456387i −0.973616 0.228193i $$-0.926718\pi$$
0.973616 0.228193i $$-0.0732819\pi$$
$$942$$ 0 0
$$943$$ −96.0000 −3.12619
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 8.00000i − 0.259965i −0.991516 0.129983i $$-0.958508\pi$$
0.991516 0.129983i $$-0.0414921\pi$$
$$948$$ 0 0
$$949$$ 4.00000i 0.129845i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −20.0000 −0.647864 −0.323932 0.946080i $$-0.605005\pi$$
−0.323932 + 0.946080i $$0.605005\pi$$
$$954$$ 0 0
$$955$$ 16.0000i 0.517748i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −8.00000 −0.258333
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 4.00000i 0.128765i
$$966$$ 0 0
$$967$$ −22.0000 −0.707472 −0.353736 0.935345i $$-0.615089\pi$$
−0.353736 + 0.935345i $$0.615089\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 28.0000i 0.898563i 0.893390 + 0.449281i $$0.148320\pi$$
−0.893390 + 0.449281i $$0.851680\pi$$
$$972$$ 0 0
$$973$$ − 16.0000i − 0.512936i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −12.0000 −0.383914 −0.191957 0.981403i $$-0.561483\pi$$
−0.191957 + 0.981403i $$0.561483\pi$$
$$978$$ 0 0
$$979$$ 32.0000i 1.02272i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −16.0000 −0.510321 −0.255160 0.966899i $$-0.582128\pi$$
−0.255160 + 0.966899i $$0.582128\pi$$
$$984$$ 0 0
$$985$$ 52.0000 1.65686
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 96.0000i − 3.05262i
$$990$$ 0 0
$$991$$ 58.0000 1.84243 0.921215 0.389053i $$-0.127198\pi$$
0.921215 + 0.389053i $$0.127198\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 20.0000i − 0.634043i
$$996$$ 0 0
$$997$$ − 18.0000i − 0.570066i −0.958518 0.285033i $$-0.907995\pi$$
0.958518 0.285033i $$-0.0920045\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2304.2.d.e.1153.1 2
3.2 odd 2 2304.2.d.g.1153.2 2
4.3 odd 2 2304.2.d.n.1153.1 2
8.3 odd 2 2304.2.d.n.1153.2 2
8.5 even 2 inner 2304.2.d.e.1153.2 2
12.11 even 2 2304.2.d.p.1153.2 2
16.3 odd 4 1152.2.a.e.1.1 yes 1
16.5 even 4 1152.2.a.p.1.1 yes 1
16.11 odd 4 1152.2.a.n.1.1 yes 1
16.13 even 4 1152.2.a.g.1.1 yes 1
24.5 odd 2 2304.2.d.g.1153.1 2
24.11 even 2 2304.2.d.p.1153.1 2
48.5 odd 4 1152.2.a.f.1.1 yes 1
48.11 even 4 1152.2.a.d.1.1 1
48.29 odd 4 1152.2.a.q.1.1 yes 1
48.35 even 4 1152.2.a.o.1.1 yes 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.a.d.1.1 1 48.11 even 4
1152.2.a.e.1.1 yes 1 16.3 odd 4
1152.2.a.f.1.1 yes 1 48.5 odd 4
1152.2.a.g.1.1 yes 1 16.13 even 4
1152.2.a.n.1.1 yes 1 16.11 odd 4
1152.2.a.o.1.1 yes 1 48.35 even 4
1152.2.a.p.1.1 yes 1 16.5 even 4
1152.2.a.q.1.1 yes 1 48.29 odd 4
2304.2.d.e.1153.1 2 1.1 even 1 trivial
2304.2.d.e.1153.2 2 8.5 even 2 inner
2304.2.d.g.1153.1 2 24.5 odd 2
2304.2.d.g.1153.2 2 3.2 odd 2
2304.2.d.n.1153.1 2 4.3 odd 2
2304.2.d.n.1153.2 2 8.3 odd 2
2304.2.d.p.1153.1 2 24.11 even 2
2304.2.d.p.1153.2 2 12.11 even 2